Theorem xpsneng 4422

Description: A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254.

Assertion

Ref	Expression
xpsneng	|- ((A e. C /\ B e. D) -> (A X. {B}) ~~ A)

Proof of Theorem xpsneng

Step	Hyp	Ref	Expression
1		xpeq1 3195	. . 3 |- (x = A -> (x X. {y}) = (A X. {y}))
2		id 59	. . 3 |- (x = A -> x = A)
3	1, 2	breq12d 2626	. 2 |- (x = A -> ((x X. {y}) ~~ x <-> (A X. {y}) ~~ A))
4		sneq 2413	. . . 4 |- (y = B -> {y} = {B})
5		xpeq2 3196	. . . 4 |- ({y} = {B} -> (A X. {y}) = (A X. {B}))
6	4, 5	syl 10	. . 3 |- (y = B -> (A X. {y}) = (A X. {B}))
7	6	breq1d 2624	. 2 |- (y = B -> ((A X. {y}) ~~ A <-> (A X. {B}) ~~ A))
8		visset 1809	. . 3 |- x e. V
9		visset 1809	. . 3 |- y e. V
10	8, 9	xpsnen 4421	. 2 |- (x X. {y}) ~~ x
11	3, 7, 10	vtocl2g 1846	1 |- ((A e. C /\ B e. D) -> (A X. {B}) ~~ A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  {csn 2405   class class class wbr 2614   X. cxp 3163   ~~ cen 4354

This theorem is referenced by:  cdafi 4916

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-en 4357

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